Newspace parameters
| Level: | \( N \) | \(=\) | \( 475 = 5^{2} \cdot 19 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 475.a (trivial) |
Newform invariants
| Self dual: | yes |
| Analytic conductor: | \(3.79289409601\) |
| Analytic rank: | \(1\) |
| Dimension: | \(3\) |
| Coefficient field: | 3.3.169.1 |
|
comment: defining polynomial
gp: f.mod \\ as an extension of the character field
|
|
| Defining polynomial: |
\( x^{3} - x^{2} - 4x - 1 \)
|
| Coefficient ring: | \(\Z[a_1, a_2]\) |
| Coefficient ring index: | \( 1 \) |
| Twist minimal: | yes |
| Fricke sign: | \(1\) |
| Sato-Tate group: | $\mathrm{SU}(2)$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring in terms of a root \(\nu\) of
\( x^{3} - x^{2} - 4x - 1 \)
:
| \(\beta_{1}\) | \(=\) |
\( \nu \)
|
| \(\beta_{2}\) | \(=\) |
\( \nu^{2} - \nu - 3 \)
|
| \(\nu\) | \(=\) |
\( \beta_1 \)
|
| \(\nu^{2}\) | \(=\) |
\( \beta_{2} + \beta _1 + 3 \)
|
Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | |||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1.1 |
|
−2.37720 | −1.27389 | 3.65109 | 0 | 3.02830 | −0.726109 | −3.92498 | −1.37720 | 0 | |||||||||||||||||||||||||||
| 1.2 | −1.27389 | 1.65109 | −0.377203 | 0 | −2.10331 | −3.65109 | 3.02830 | −0.273891 | 0 | ||||||||||||||||||||||||||||
| 1.3 | 1.65109 | −2.37720 | 0.726109 | 0 | −3.92498 | 0.377203 | −2.10331 | 2.65109 | 0 | ||||||||||||||||||||||||||||
Atkin-Lehner signs
| \( p \) | Sign |
|---|---|
| \(5\) | \(-1\) |
| \(19\) | \(-1\) |
Inner twists
This newform does not admit any (nontrivial) inner twists.
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 475.2.a.e | ✓ | 3 |
| 3.b | odd | 2 | 1 | 4275.2.a.bm | 3 | ||
| 4.b | odd | 2 | 1 | 7600.2.a.cc | 3 | ||
| 5.b | even | 2 | 1 | 475.2.a.g | yes | 3 | |
| 5.c | odd | 4 | 2 | 475.2.b.b | 6 | ||
| 15.d | odd | 2 | 1 | 4275.2.a.ba | 3 | ||
| 19.b | odd | 2 | 1 | 9025.2.a.bc | 3 | ||
| 20.d | odd | 2 | 1 | 7600.2.a.bh | 3 | ||
| 95.d | odd | 2 | 1 | 9025.2.a.y | 3 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 475.2.a.e | ✓ | 3 | 1.a | even | 1 | 1 | trivial |
| 475.2.a.g | yes | 3 | 5.b | even | 2 | 1 | |
| 475.2.b.b | 6 | 5.c | odd | 4 | 2 | ||
| 4275.2.a.ba | 3 | 15.d | odd | 2 | 1 | ||
| 4275.2.a.bm | 3 | 3.b | odd | 2 | 1 | ||
| 7600.2.a.bh | 3 | 20.d | odd | 2 | 1 | ||
| 7600.2.a.cc | 3 | 4.b | odd | 2 | 1 | ||
| 9025.2.a.y | 3 | 95.d | odd | 2 | 1 | ||
| 9025.2.a.bc | 3 | 19.b | odd | 2 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{3} + 2T_{2}^{2} - 3T_{2} - 5 \)
acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(475))\).
Hecke characteristic polynomials
$p$
$F_p(T)$
$2$
\( T^{3} + 2 T^{2} - 3 T - 5 \)
$3$
\( T^{3} + 2 T^{2} - 3 T - 5 \)
$5$
\( T^{3} \)
$7$
\( T^{3} + 4T^{2} + T - 1 \)
$11$
\( T^{3} - T^{2} - 4T - 1 \)
$13$
\( T^{3} + 3 T^{2} - 36 T - 103 \)
$17$
\( T^{3} + 14 T^{2} + 61 T + 79 \)
$19$
\( (T - 1)^{3} \)
$23$
\( T^{3} + 8 T^{2} - 9 T - 125 \)
$29$
\( T^{3} + 5 T^{2} + 4 T - 5 \)
$31$
\( T^{3} + T^{2} - 30 T + 53 \)
$37$
\( T^{3} + 5 T^{2} - 74 T - 395 \)
$41$
\( T^{3} - T^{2} - 82 T + 155 \)
$43$
\( T^{3} - 5 T^{2} - 113 T + 317 \)
$47$
\( T^{3} + 9 T^{2} - 64 T - 311 \)
$53$
\( T^{3} + 31 T^{2} + 303 T + 905 \)
$59$
\( T^{3} + 6 T^{2} - 40 T - 200 \)
$61$
\( T^{3} - 3 T^{2} - 10 T - 1 \)
$67$
\( T^{3} - 13T^{2} + 169 \)
$71$
\( T^{3} - 7T^{2} - T + 47 \)
$73$
\( T^{3} + T^{2} - 30 T + 53 \)
$79$
\( T^{3} + 18 T^{2} + 17 T - 395 \)
$83$
\( T^{3} + 3 T^{2} - 270 T + 131 \)
$89$
\( T^{3} + 20 T^{2} + 51 T - 125 \)
$97$
\( T^{3} - 13T^{2} + 169 \)
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